Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

Abstract

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), Rem 3.15]) from the finite generation thm (Birkar-Cascini-Hacon-McKernan [BCHM]) of the lc rings for klt pairs and from the fact (cf. Mourougane-Russo [MoRu, C.R.A.S. Math. 325 (1997)]) that a nef and abundant $\mathbf{Q}$-divisor $D$ is semiample if its graded ring is finitely generated:"For a nef and abundant lc divisor which is klt, the lc ring is finitely generated, thus it is semiample." [BCHM] first proved that the minimal model program runs for big klt lc divisors and next implied the finite generation of the lc rings for klt lc divisors which are not necessarily big from the Fujino-Mori lc bdle formula ([FM, J. Differential Geom., 56 (2000)]). Mourougane-Russo [MoRu] implies the semiampleness of a nef and abundant $\mathbf{Q}$-divisor whose graded ring is finitely generated, using the Kawamata numerically trivial fibrations ([Kawamata. Pluricanonical systems. Invent. Math. 79 (1985)]). Consequently the author withdraw the article.